
jointly ergodic for the system \((X,\mu , T)\), if for all functions \(f_1,\ldots , f_\ell \in L^\infty (\mu )\)  we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_\ell (n)}f_\ell = \int f_1\, d\mu \cdot \ldots \cdot \int f_\ell \, d\mu \) 
where convergence takes place in \(L^2(\mu )\) .This should not be confused with the notion of joint ergodicity introduced in [1]}, which is a property of a collection of measure preserving transformations and not a property of a collection of sequences.

jointly ergodic, if it is jointly ergodic for every ergodic system.


